A256907 -id:A256907 - cp's OEIS frontend

A256937 Numbers n such that phi(n) = 4*phi(n+1).

629, 1469, 85139, 100889, 139859, 154979, 168149, 304079, 396899, 838199, 1107413, 1323449, 1465463, 2088839, 2160899, 2504879, 2684879, 2693249, 2800181, 3404609, 3512249, 3576869, 3885881, 4241819, 4500509, 4620659, 4822649, 5530709, 5805449

Offset: 1

Mauro Fiorentini, Apr 13 2015

nonn

			phi(629) = 576 = 4*phi(630).

Mauro Fiorentini, Table of n, a(n) for n = 1..154 (all terms for n up to 10^9).

Cf. A171262, A256907.

[n: n in [1..10^7] | EulerPhi(n) eq 4*EulerPhi(n+1)]; // Vincenzo Librandi, Apr 14 2015

A:= NULL:
y:= numtheory:-phi(1):
for n from 1 to 10^6 do
x:= numtheory:-phi(n+1);
if y = 4*x then A:= A, n fi;
y:= x;
od:
A;  # Robert Israel, Apr 15 2015

Select[Range@ 1000000, EulerPhi@ # == 4 EulerPhi[# + 1] &] (* Michael De Vlieger, Apr 13 2015 *)
Position[Partition[EulerPhi[Range[6*10^6]],2,1],?(#[[1]]==4#[[2]]&),{1},Heads->False]//Flatten (* _Harvey P. Dale, Sep 18 2016 *)

s=[]; for(n=1, 1000000, if(eulerphi(n)==4*eulerphi(n+1), s=concat(s, n))); s \\ Colin Barker, Apr 13 2015

[n for n in (1..1000000) if euler_phi(n) == 4*euler_phi(n+1)]; # Bruno Berselli, Apr 14 2015

A257550 Numbers n such that phi(n) = 5*phi(n+1).

Original entry on oeis.org

17907119, 18828809, 31692569, 73421039, 179467469, 322757819, 337567229, 627702389, 975314339, 2537636009, 2722271369, 3328653509, 3917646809, 5529412349, 6369847469, 11179199849, 11201693579, 11363832479, 13442120999, 16781760449, 19751331599, 20002320029

Offset: 1

Ray Chandler, Apr 29 2015

nonn

			phi(17907119) = 16588800 = 5*phi(17907120).

Giovanni Resta, Table of n, a(n) for n = 1..103 (terms < 10^12)

Cf. A001274, A171262, A256907, A256937.

a1={};nmax=10^9;last=EulerPhi[1];n=2;
While[nRay Chandler, Apr 30 2015 *)

a(10)-a(22) from Giovanni Resta, May 11 2015

A257865 Smallest k such that phi(k) = n*phi(k+1), where phi(n) = A000010(n) gives the value of Euler's totient function at n.

Original entry on oeis.org

1, 5, 119, 629, 17907119

Offset: 1

Felix Fröhlich, May 11 2015

From Manfred Scheucher, May 27 2015: (Start)
a(6)>=3*10^8 (calculation)
a(7)>=3.5*10^13, a(8)>=4.5*10^25, a(9)>=3.0*10^47, and so on... (doubly exponential lower bound, see uploaded pdf)
239719159679 and 239742643139 admit a ratio of 5.998... and 6.008..., resp.
There might be a relation to the sequence A098026. (End)

			a(3) = 119, because phi(119) == 3*phi(120) = 96 and 119 is the smallest k where this equality holds for n = 3.

Manfred Scheucher, A lower bound on A257865(n)

Cf. A001274, A171262, A256907, A256937, A257550.

Table[k = 1; While[EulerPhi[k] != n EulerPhi[k + 1], k++]; k, {n, 4}] (* Michael De Vlieger, May 12 2015 *)

a(n) = my(k=1); while(eulerphi(k)!=n*eulerphi(k+1), k++); k

a(n) >= exp(exp(c(n-3))) with c=exp(gamma) and gamma being the Euler-Mascheroni_constant (see pdf). - Manfred Scheucher, May 27 2015

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A256937 Numbers n such that phi(n) = 4*phi(n+1).

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A257550 Numbers n such that phi(n) = 5*phi(n+1).

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A257865 Smallest k such that phi(k) = n*phi(k+1), where phi(n) = A000010(n) gives the value of Euler's totient function at n.

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